Let $A = \{1, 2, 3\}, B = \{1, 3, 5\}$. $A$ relation $R: A \to B$ is defined by $R = \{(1, 3), (1, 5), (2, 1)\}$. Then ${R^{-1}}$ is:

Suppose $f(x) = (x + 1)^2$ for $x \ge -1$. If $g(x)$ is the function whose graph is the reflection of the graph of $f(x)$ with respect to the line $y = x$,then $g(x)$ equals

If $f(x) = (x+1)^2 - 1$ for $x \geq -1$,then find the set $\{x \mid f(x) = f^{-1}(x)\}$.

Consider $f: \{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g: \{a, b, c\} \rightarrow \{\text{apple, ball, cat}\}$ defined as $f(1)=a, f(2)=b, f(3)=c$ and $g(a)=\text{apple}, g(b)=\text{ball}, g(c)=\text{cat}$. Show that $f, g$ and $g \circ f$ are invertible. Find $f^{-1}, g^{-1}$ and $(g \circ f)^{-1}$ and show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Let $f: R \rightarrow R$ be defined as $f(x)=10x+7$. Find the function $g: R \rightarrow R$ such that $g \circ f = f \circ g = I_{R}$.

If $f: R \rightarrow R$ is defined by $f(x)=x^{3}$,then $f^{-1}(8)$ is equal to